2  Population Means (8.2.1 - 8.2.2)

When we take a random sample of size \(n\) from a normal population with unknown mean, say \(X\sim N(\mu, \sigma)\), it is often of interest to use the sample \(x_1, x_2, \ldots , x_n\) with sample mean \(\bar{x}\) to estimate the value of the population mean \(\mu\). Confidence intervals can be used to find a range of plausible values that \(\mu\) could be, based on the random sample.

It is important to pay close attention to the variance of the underlying distribution and ask, do we know what it is? The answer to this question will change the way in which the confidence interval is calculated.

2.1 Variance is known (8.2.1)

When the variance of the underlying normal distribution is known, then a \((1-\alpha)\cdot100\%\) confidence interval for the population mean, \(\mu\), is given by,

\[\begin{equation} \tag{8.5} CI_{1-\alpha}(\mu)=\left[\bar x-z_{1-\frac{\alpha}{2}}\cdot\frac{\sigma}{\sqrt{n}},\,\,\bar x+z_{1-\frac{\alpha}{2}}\cdot\frac{\sigma}{\sqrt{n}}\right] \end{equation}\]

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See Section 8.2.1 of Probability and Statistics with R for some examples of using this result.

2.2 Variance is unknown (8.2.2)

If the variance of the normal distribution that the random sample is taken from is unknown, then a \((1-\alpha)\cdot100\%\) confidence interval for the population mean, \(\mu\), is given by,

\[\begin{equation} \tag{8.10} CI_{1-\alpha}(\mu)=\left[\bar x-t_{1-\frac{\alpha}{2};n-1}\cdot\frac{s}{\sqrt{n}},\,\,\bar x+t_{1-\frac{\alpha}{2};n-1}\cdot\frac{s}{\sqrt{n}}\right] \end{equation}\]

• \(s\) is the sample standard deviation. The sample variance can be found using, \[s^2=\frac{\sum_{i=1}^n(x_i-\bar{x})^2}{n-1}=\frac{\sum_{i=1}^nx_i^2-n\bar x^2}{n-1}\]

  Remember to take the square root of the sample variance to find the sample standard deviation.

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See Section 8.2.2 of Probability and Statistics with R for the derivation of this result and some examples of using it in practise.